Skip to main content
SpringerLink
Log in

Spectra of three-dimensional cruciform and lattice quantum waveguides

  • Mathematical Physics
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

It is shown that the discrete spectrum of the Dirichlet problem for the Laplacian on the union of two mutually perpendicular circular cylinders consists of a single eigenvalue, while the homogeneous problem with a threshold value of the spectral parameter has no bounded solutions. As a consequence, an adequate one-dimensional model of a square lattice of thin quantum waveguides is presented and the asymptotic behavior of the spectral bands and lacunas (zones of wave transmission and deceleration) and the oscillatory processes they generate is described.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Exner and O. Post, J. Geom. Phys. 54 (1), 77–115 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Kuchment and O. Post, Commun. Math. Phys. 275 (3), 805–826 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Grieser, Proc. London Math. Soc. 97 (3), 718–752 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  4. L. Pauling, J. Chem. Phys. 4 (10), 673–677 (1936).

    Article  Google Scholar 

  5. S. A. Nazarov, Probl. Mat. Anal. 73, 101–127 (2013).

    Google Scholar 

  6. S. A. Nazarov, Comput. Math. Math. Phys. 54 (8), 1261–1279 (2014).

    Article  MathSciNet  Google Scholar 

  7. S. A. Nazarov, Dokl. Math. 90 (2), 637–641 (2014).

    Article  MATH  Google Scholar 

  8. S. A. Nazarov, St. Petersburg Math. J. 23 (2), 351–379 (2011).

    Article  Google Scholar 

  9. M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningrad. Gos. Univ., Leningrad, 1980; Reidel, New York, 1986).

    Book  Google Scholar 

  10. G. Pólya and G. Szegó, Isoperimetric Inequalities in Mathematical Physics (Princeton Univ. Press, Princeton, N.J., 1951).

    MATH  Google Scholar 

  11. M. M. Skriganov, Proc. Steklov Inst. Math. 171, 1–121 (1987).

    MathSciNet  Google Scholar 

  12. S. A. Nazarov and A. V. Shanin, Appl. Anal. 93 (3), 572–582 (2014).

    Article  MATH  MathSciNet  Google Scholar 

  13. L. Friedlander and M. Solomyak, Israel J. Math. 170, 337–354 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  14. S. A. Nazarov, Theor. Math. Phys. 167 (2), 606–627 (2011).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, Universitetskii pr. 28, Staryi Peterhof, St. Petersburg, 199504, Russia

    F. L. Bakharev, S. G. Matveenko & S. A. Nazarov

  2. St. Petersburg Branch, National Research University “Higher School of Economics,”, ul. Sedova 55, korp. 2, St. Petersburg, 193171, Russia

    S. G. Matveenko

  3. St. Petersburg State Polytechnical University, ul. Politekhnicheskaya 29, St. Petersburg, 195251, Russia

    S. A. Nazarov

  4. Institute of Mechanical Engineering Problems, Russian Academy of Sciences, Vasil’evskii Ostrov, Bol’shoi pr. 61, St. Petersburg, 199178, Russia

    S. A. Nazarov

Authors
  1. F. L. Bakharev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. G. Matveenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. A. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. L. Bakharev.

Additional information

Original Russian Text © F.L. Bakharev, S.G. Matveenko, S.A. Nazarov, 2015, published in Doklady Akademii Nauk, 2015, Vol. 463, No. 6, pp. 650–654.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bakharev, F.L., Matveenko, S.G. & Nazarov, S.A. Spectra of three-dimensional cruciform and lattice quantum waveguides. Dokl. Math. 92, 514–518 (2015). https://doi.org/10.1134/S1064562415040274

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562415040274

Keywords

Search

Navigation